Papers

We give an example of a finite ring for which the set of polynomials inducing the zero function fails to be a two-sided ideal, disproving a conjecture of Werner.

Fluid antenna array (FAA) activation jointly determines the effective multi-user channel for precoding and the sparse physical aperture. Channel-oriented selection can concentrate high-gain ports and erode aperture quality, whereas geometry-oriented selection does not adapt to instantaneous channel state information (CSI). This paper formulates finite-port FAA activation as a rate–aperture–feasibility problem under an exact RF-chain budget. We propose impedance-aware zonal port activation (IA-ZPA), which couples compact CSI-conditioned port scoring with a checkerboard feasibility projection and inference-time mutual-impedance-aware selection. The learned scorer ranks ports, while the deterministic rule fixes the active aperture; a separate current-domain RZF backend then evaluates source-drive feasibility. Under a common induced-EMF protocol, IA-ZPA attains the largest constrained rate among the methods satisfying the prescribed mean-PSLL target with a substantially lower decision time than greedy selection.

We study pull-based remote state estimation of an arbitrary, multi-state Markov source while accounting for both freshness and correctness attributes of information. To that end, we formulate a discounted optimization problem in terms of the age of incorrect information (AoII), and express it as a joint source-AoII belief Markov decision process (MDP) under maximum a posteriori (MAP) estimation. We then exploit the information structure of the model and prove that every reachable belief is represented by the last successfully observed source state and the number of time slots elapsed since that observation. For numerical computation, we truncate the elapsed no-success duration at a finite level and derive an explicit error bound and a criterion for selecting the truncation parameter. For reliable links, we show that an optimal policy can be represented by a look-up table of waiting times. For unreliable links, we propose a persistent policy and derive computable performance bounds. We also show that the MAP estimate stabilizes after a finite number of time slots. To further reduce memory requirements, we introduce a hybrid estimator with an early stationary switch and derive a computable bound on the resulting difference in performance. Finally, we extend the framework to multiple sources, formulate the scheduling problem as a restless multi-armed bandit, establish a sufficient condition for indexability, and develop an approximate Whittle index policy based on interpolation. Our numerical results illustrate the structure of the optimal single-source policy, evaluate the performance of the multi-source policies, and verify that the proposed heuristic policies closely approach the optimal solution while substantially reducing computational efforts.

Chapoton introduced an interesting family of polytopes called arbor polytopes, where a polytope $\mathcal{Q}_\tau$ is associated to any arbor $\tau$. Chapoton conjectured that the polynomial $h(\tau)$ which counts lattice points in $\mathcal{Q}_\tau$ by number of nonzero entries is palindromic and unimodal. Athanasiadis, Xiao, and Yan recently proved that $h(\tau)$ is gamma-positive for a certain class of arbors they call octopuses. Since their proof was computational, they asked for one that is bijective. We present such a proof. We also extend their results by showing that $h(\tau)$ is gamma-positive for a larger class of arbors we call lopsided octopuses.

We study integer-valued polynomials and null polynomials over structural matrix rings, that is, rings whose elements are matrices in which an entry may be nonzero only when its row index precedes its column index in a fixed preorder. We prove that the integer-valued polynomials over a structural matrix ring with entries in an integral domain form a ring, and that the null polynomials over a structural matrix ring with entries in an arbitrary commutative ring form a two-sided ideal. In both settings, we give a characterization of the corresponding polynomials with matrix coefficients in terms of scalar-coefficient polynomials. These results extend corresponding theorems for full matrix rings and upper triangular matrix rings.

We construct a finite state sum from a link diagram labelled by transpositions. The diagram presents a closed 3-manifold as a four-fold simple branched cover, and the state sum equals the SU(2)-type Turaev-Viro invariant of that manifold. For each admissible colouring of the diagram's edges and complementary regions, a global factor is combined with crossing weights obtained by summing over compatible local colourings. We then sum these products over all diagram colourings. The formula is proved by assembling crossing-wise local triangulations and matching the two state sums.

In this paper, we study the random Laguerre tessellation, a weighted generalization of the Voronoi tessellation, generated by a general stationary marked point process. We first derive a nearly optimal sufficient condition on the generating marked point process which ensures that the resulting random Laguerre tessellation is well-defined, utilising the concept of tempered configurations to handle potentially unbounded weights. We then investigate how the three mixing properties - ergodicity, mixing and $\alpha$-mixing - of the generating marked point process are preserved for the corresponding random Laguerre tessellation. Our approach combines standard approximation arguments with the properties of tempered configurations and the measurability of the Laguerre mapping.

The spectral side of the Kuznetsov trace formula for $GL(2)$ is governed by the Bessel-Kuznetsov integral transform $\check{\phi}(t)$. While classical bounds guarantee rapid decay of this transform for smooth, non-oscillatory test functions, modern applications in analytic number theory—particularly those involving twisted shifted convolution sums—frequently encounter test functions exhibiting a highly oscillatory linear phase $e(\alpha x)$. In this paper, we provide a rigorous and explicit asymptotic analysis of $\check{\phi}(t)$ in the semiclassical limit $t \to \infty$ under such oscillatory conditions. By applying the WKB approximation to the imaginary-order Bessel kernel, we identify a sharp phase transition dependent on the twist parameter $\alpha$. We prove that in the sub-critical regime ($\alpha \le 1/2\pi$), the transform decays rapidly. Conversely, in the super-critical regime ($\alpha > 1/2\pi$), the geometric oscillations resonate with the spectral kernel, yielding a localized main term of order $O(t^{-1})$ with a remarkably simplified arithmetic phase.

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $\sigma$-centered forcing notions gives a model of $ MA_{\omega_1}(\sigma$-centered$)+2^{\aleph_0}=\aleph_2 +\omega_1^2\nrightarrow(\omega_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $\omega_1$ are countable unions of linked sets. Under $\text{MA}_{\omega_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.

Kinetically constrained lattice gases are interacting particle systems with conserved number of particles, having degenerate rates caused by a kinetic constraint. Over the past decade, there has been major progress in the study of their non-conservative counterpart, kinetically constrained spin models: in dimensions 1 and 2, we have a good understanding of the universality classes of these models. However, the conservative systems are much more challenging to analyze, and very few results are available. The purpose of this paper is to describe universality in the one dimensional case. We will characterize the three universality classes, determining whether such a model is always ergodic, never ergodic, or exhibits an ergodicity phase transition.

In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of $f$-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.

This paper provides a complete analytical study of the positive stabilization, positive state estimation, and observer-based positive stabilization problems for a generic boundary control diffusion system with point observation. In particular, optimal design problems are solved analytically, and an original strategy for observer-based feedback stabilization is proposed. The three problems are also investigated for an arbitrary-order spatial discretization of the nominal PDE system enabling numerical implementation together with rigorous comparisons with the nominal solutions. In both the continuous and discretized settings, explicit and readily implementable solutions are derived.

For each $r\in\{4,6\}$, we construct an explicit one-parameter family of elliptic curves over $\mathbb{Q}(i)$ containing infinitely many pairwise nonisomorphic curves genuinely defined over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $r$. We construct explicit $\mathbb{Q}(i)$-rational points to bound the ranks from below. Kai's theorem on prime values of linear patterns over number fields provides specializations with controlled local behavior, allowing us to obtain matching upper bounds via $[1+i]$-descent. The construction extends the strategy of the author's earlier rank-$2$ paper by replacing a symmetric Gaussian-prime configuration with systems of binary linear forms satisfying several complementary square identities. In the rank-$6$ case, the support vectors attached to the three constructed points and $(0,0)$ span the self-dual Reed-Muller code $\mathrm{RM}(1,3)$, which also occurs as the kernel of the quadratic-residue Laplacian governing the Selmer group.

We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization – the implicit-function sensitivity of the entropic OT plan to the cost – together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent. The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation. T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway–Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

Let $\Omega \subset \mathbb{C}^{n}$ and $\xi \in \ell^{1}$. The $\xi$-Bergman kernel $K_{\xi, \Omega}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $\xi$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $\xi$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $\xi$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in \Omega$ satisfies \[ B_{\xi,\Omega}(z,z)=K_{\xi,\Omega}(z), \] thereby recovering the $\xi$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $\xi$-Bergman kernel is real analytic on $\Omega$. We also establish biholomorphic transformation laws, and obtain a representation of the $\xi$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $\xi$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $\xi$, establish corresponding $\xi$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.

In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congruences for $g_n(x)$ and $v_n(x)$ given by \begin{align*} &g_0(x)=1,\ g_1(x)=\frac{x+1}2,\ (n+1)^2g_{n+1}(x)=\Big(2n(n+1)+\frac{x+1}2\Big)g_n(x)-n^2g_{n-1}(x)\ (n\ge 1), \&v_0(x)=1,\ v_1(x)=x,\ (n+1)^3v_{n+1}(x)=(2n+1)(n(n+1)+x)v_n(x)-n^3v_{n-1}(x)\ (n\ge 1).\end{align*}

Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D—that is, Ninomiya's adaptive 9-point Newton–Cotes rule extended with a sequence of higher-order rules—and Takahasi and Moris' double exponential (DE) formula. We evaluate them for Kahaner's 21 test problems. For both absolute tolerances $10^{-50}$ and $10^{-100}$, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as $1/\sqrt{x}$, it requires about $5.4\times10^{7}$ function evaluations at $10^{-100}$, roughly $7\times10^{4}$ times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.

We study Toeplitz determinants $\det T_n(e^f)$ for $f$ whose Fourier coefficients satisfy $f_k=O(|k|^{-1})$. This regime extends beyond $H^{1/2}$ and includes symbols with Fisher-Hartwig singularities. We develop an operator-theoretic approach based on the Baker-Campbell-Hausdorff formula that separates the quadratic term \[ \sum_{k=1}^{\infty}\min(k,n)f_kf_{-k} \] from the higher-order terms in the expansion of $\log\det T_n(e^{tf})$. We show that this quadratic term accounts for the possible growth with $n$, while every fixed higher-order coefficient remains bounded. For symbols with bounded positive and negative Fourier parts, our estimates yield two-sided bounds for the determinant after removal of the quadratic contribution. For a broader admissible class, including Fisher-Hartwig-type symbols, we obtain uniform higher-order coefficient bounds and a central limit theorem for the associated CUE linear statistics. We also obtain bounds on mixed exponential moments for CUE-derived random fields beyond the characteristic polynomial.

The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero–nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.

A set in $\mathbb{R}^d$ is $s$-convex if it is the union of at most $s$ convex sets. A family $F$ satisfies the $(p,q)$ property if among any $p$ sets in $F$, some $q$ intersect. Let $\mathrm{HD}_d^{(s)}(p,q)$ be the minimum number of points needed to pierce a finite family of $s$-convex sets that satisfies the $(p,q)$-property. Alon and Kalai (1995) proved that $\mathrm{HD}_d^{(s)}(p,q)$ exists for any $p \geq q \geq d+1$ and any $s \geq 1$, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general $d$ and for $s$-intervals of the line (i.e., $\mathrm{HD}_1^{(s)}(p,q)$). In particular, we prove the following: (i) For every $d\ge2$, $s \geq 1$ and $\delta>0$, if $p>q$ and $q\ge C_d\log(e sp)$, then $\mathrm{HD}_d^{(s)}(p,q) \le p-q+1 + O_{d,\delta}((s \cdot \tfrac{p}{q} \cdot \log \tfrac{esp}{q})^{\rho_d+\delta}),$ where $\rho_d<d$ is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For $s \geq 1$, $p \geq q \geq 2$ and $q\ge C_0s\log(2s)\log(ep)$, $p-q+s \leq \mathrm{HD}_1^{(s)}(p,q) \leq p-q+2s+1$. This result provides the first near-tight estimate for $\mathrm{HD}_1^{(s)}(p,q)$ for $q>2$. (iii) For any fixed $s$, there are an integer $\kappa_s\in\{s,\ldots,2s\}$ and constants $C_s,p_s>0$ such that, whenever $p\ge p_s$ and $q\ge C_s\log(ep)$, $ \mathrm{HD}_1^{(s)}(p,q)\in\{p-q+\kappa_s,\;p-q+\kappa_s+1\}. $ Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any $s \geq 1$, $\mathrm{HD}_3^{(s)}(p,4) \geq sp^{2-o(1)}$. Already for families of convex sets, this significantly improves the best known lower bound on $\mathrm{HD}_d^{(1)}(p,d+1)$, for all $d \geq 3$.

We extend the notion of $\mathbf{g}$-vector fan so that it is defined for a Hom-finite Krull-Schmidt 0-Auslander $k$-linear extriangulated category $\mathcal{C}$ with a projective silting object $T$. Moreover, we show that the $\mathbf{g}$-vector fan admits an admissible partition, in the sense of the second-named author, which is induced by thick subcategories. One can thus define the picture category of $\mathcal{C}$. We establish a bijection between thick subcategories of $\mathcal{C}$ generated by presilting objects containing all projective-injective objects and $\tau$-perpendicular subcategories of the endomorphism $k$-algebra of $T$. This shows that our construction unifies all previous constructions of picture categories and $\tau$-cluster morphism categories of finite-dimensional algebras. We introduce morphisms of partitioned fans to provide a common framework for the functorial relationships between picture categories of different algebras and categories.

In this paper, we establish the $\mathcal{B}^4$-almost periodicity in the sense of Besicovitch for a suitably normalized error term associated with a broad class of arithmetical functions introduced by Chandrasekharan and Narasimhan. This result significantly strengthens $\mathcal{B}^2$-almost periodicity previously investigated in the literature for related error terms. By deriving truncated Voronoï-type formulas, we demonstrate that the normalized error term lies in the Besicovitch space $B^4$. As a consequence, we deduce that the error term admits a limit probability distribution and establish an explicit formula for its mean fourth power moment.

We present two counterexamples to the Markovity Conjecture of Gohari, Liu and Nair (ISIT 2025), which is a structural conjecture concerning the optimizers of the dual functional associated with Marton's inner bound and, if true, would greatly simplify the evaluation of Marton's inner bound. Both counterexamples are ternary-input broadcast channels with strictly positive transition probabilities and use the same nonrectangular $2\times2$ auxiliary structure. In each case, we exhibit an explicit non-Markov construction whose objective value is rigorously larger than that achievable by any construction satisfying the conjectured Markov structure. Both examples are obtained with the assistance of GPT-5.6 Sol and disprove the Markovity Conjecture.

We are concerned with the global classical solution problem of 3D compressible isentropic Euler equations of Chaplygin gases \[ \begin{cases} \partial_t\rho + \mathrm{div}(\rho v) = 0,\\ \partial_t(\rho v) + \mathrm{div}(\rho v \otimes v) + \nabla p = 0,\\ \rho(0,x) = \bar\rho + \varepsilon\rho_0(x),\ v(0,x) = \varepsilon v_0(x). \end{cases} \] where $\bar\rho>0$ is a constant, $\varepsilon>0$ is small, the state equation is $p=p(\rho)=P_0-\frac{D}{\rho}$ with $P_0$ and $D$ being some positive constants. For the 3D compressible Euler equations of Chaplygin gases, which are a prototype of multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues, there is a basic conjecture imposed by A. Majda: it typically has a global classical solution $(\rho, v)$ with $(\rho-\bar\rho, v)\in C([0,\infty), H^s(\Bbb R^3)) \cap C^1([0,\infty), H^{s-1}(\Bbb R^3))$ when $(\rho_0, v_0)\in H^s(\Bbb R^3)$ with $s>\frac52$ unless $(\rho, v)$ itself blows up in finite time. In this paper, under the assumptions that for any fixed constant $\mu$ with $0<\mu<1/2$, integer $N\geq 15$, $\mathrm{rot}\,v_0(x) \equiv 0$ and \[ \|(\rho_0, v_0)\|_{H^{N}(\mathbb{R}^3)}+\sum_{|a|\leq 13} \|\langle x\rangle^{1+\mu} \nabla^a(\rho_0,v_0)\|_{L^2(\mathbb{R}^3)} \leq 1, \] we show that the classical solution $(\rho, v)$ exists globally. Our main ingredients include: establishing a series of new decay estimates of energy bounds, weighted pointwise space-time $L^\infty$-$L^2$ estimates and weighted Strichartz-type estimates for the 3D potential flow equation of Chaplygin gases.

We introduce reparametrization categories with cuts. For every such category $\mathcal{P}$, the tensor product of two objectwise weak homotopy equivalences of $\mathcal{P}$-spaces is an objectwise weak homotopy equivalence, and the q-model structure on $\mathcal{P}$-flows is left proper. The reparametrization categories $\mathcal{G}$ and $\mathcal{M}$, as well as the terminal category $\mathbf{1}$, have cuts. The terminal category recovers the left properness of the q-model structure on ordinary flows. We also give a symmetric reparametrization category with cuts and a strict unit. This category is not equivalent to $\mathbf{1}$.

The most well-known beyond-Fourier heat conduction model, the Maxwell–Cattaneo–Vernotte equation is solved analytically in the presence of heat transfer boundary condition. In contrast to the corresponding Fourier problem, this boundary condition renders the underlying differential operator non-selfadjoint. With a suitable scalar product, the adjoint eigenvalue problem is established. The resulting left and right eigenfunctions constitute a biorthogonal system, allowing the expansion coefficients to be determined from arbitrary square-integrable initial conditions. This enables a convenient infinite-sum analytical solution, which is presented and thoroughly investigated for various values of the model parameters (including near-Fourier and highly hyperbolic regimes) and for two practically important initial conditions (equilibrium and flash pulse initiated). The spectral structure is analyzed in detail, including the occurrence of real, imaginary, and complex-conjugate eigenvalue roots, their asymptotic distribution, and their dependence on the dimensionless relaxation time and Biot number. We find good agreement with corresponding finite-difference numerical solutions. The comparison also illustrates the effects of spectral truncation (i.e., the Gibbs phenomenon) and numerical dissipation near propagating thermal-wave fronts. Completeness of the eigenfunction set is numerically demonstrated. The initial condition induced by the flash pulse is derived analytically. The results provide an analytical benchmark for non-selfadjoint hyperbolic heat conduction, unveil how boundary heat transfer and relaxation time influence the transition between Fourier-like diffusion and thermal-wave propagation, and open the possibility to find and investigate beyond-Fourier heat conduction via heat transfer in experiments and practical applications.

For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle–path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.

We study the laws of large numbers for the largest eigenvalues among all principal minors of Wishart matrices and deformed GOE matrices. We propose a new method based on identifying the deterministic sets to which the random sets formed by suitably normalized principal minors converge in Hausdorff distance, thereby reducing the original extreme-value problems to finite-dimensional convex optimization problems. We demonstrate the effectiveness of this method in regimes not covered by the existing second-moment arguments in \cite{cai2021asymptotic,hu2023extreme}. For deformed GOE matrices with fixed minor size \(k\), we determine the limit for every diagonal variance \(a>0\) and identify a phase transition at \(a=2\). Above the transition, the limiting constant satisfies an explicit recursion with no close-form expression, and the optimizers exhibit a nested hierarchical structure, thereby resolving the case left open in \cite{cai2021asymptotic}. For Wishart matrices with general sub-Gaussian entries and fixed \(k\), we characterize the limit through an entropy-constrained deterministic convex set. When the entries are standard Gaussian, we solve the resulting optimization problem explicitly and obtain the exact value of the limiting constant.